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Aggregation-Fragmentation of Clusters in the Framework of gTASEP with Attraction Interaction

Abstract

This article provides summary of some of our results, concerning a model of aggregation and fragmentation of clusters of particles obeying the stochastic discrete-time discrete-space kinetics of the generalized Totally Asymmetric Simple Exclusion Process (gTASEP) with open boundaries. The model in essence is the ordinary TASEP with backward ordered sequential update with special kinematic interaction added, i.e., it has a second modified hopping probability \(~{{p}_{m}}\) for particles in a cluster in addition to the standard hopping probability \(p\). We consider separately the two cases of attraction interaction (\(p\) < \(~{{p}_{m}}\)): (1) the limiting case of irreversible aggregation (\({{p}_{m}}\) = 1); and (2) the generic case of attraction, when \(~p\) < \(~{{p}_{m}}\) < 1 (then aggregation and fragmentation of clusters is allowed). We put special emphasis on the use of random walk theory in the study of gTASEP. It is applied to study the inter-cluster gaps time evolution, which helps to assess the properties of the nonequilibrium stationary phases of the system and the phase transitions between them. Theoretical conclusions are in agreement with the Monte Carlo simulations.

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REFERENCES

  1. S. M. Kreidenweis, M. Petters, and U. Lohmann, “100 years of progress in cloud physics, aerosols, and aerosol chemistry research,” Am. Meteorol. Soc. 59, 11.1–11.72 (2019).

  2. M. Alifierakis, K. S. Sallah, I. A. Aksay, and J. H. Prévost, “Reversible cluster aggregation and growth model for graphene suspensions,” AICh Eng. 63, 5462–5473 (2017).

    Article  Google Scholar 

  3. G. R. Born and M. J. Cross, “Effects of inorganic ions and of plasma proteins on the aggregation of blood platelets by adenosine diphosphate,” J. Physiol. 170, 397–414 (1964).

    Article  Google Scholar 

  4. M. Bourdenx, N. S. Koulakiotis, D. Sanoudou, E. Bezard, B. Dehay, and A. Tsarbopoulos, “Protein aggregation and neurodegeneration in prototypical neurodegenerative diseases: Examples of amyloidopathies, tauopathies and synucleinopathies,” Prog. Neurobiol. 155, 171–193 (2015).

    Article  Google Scholar 

  5. M. H. Lee, “On the validity of the coagulation equation and the nature of runaway growth,” Icarus 143, 74–86 (2000).

    ADS  Article  Google Scholar 

  6. J. P. Taylor, J. Hardy, and K. H. Fischbeck, “Toxic proteins in neurodegenerative disease,” Science 296, 1991 (2002).

    ADS  Article  Google Scholar 

  7. J. Krug, “Boundary-induced phase transitions in driven diffusive systems,” Phys. Rev. Lett. 67, 1882 (1991).

    ADS  Article  Google Scholar 

  8. D. Chowdhury, L. Santen, and A. Schadschneider, “Statistical physics of vehicular traffic and some related systems,” Phys. Rep. 329, 199 (2000).

    ADS  MathSciNet  Article  Google Scholar 

  9. S. Katz, J. L. Lebowitz, and H. Spohn, “Phase transitions in stationary nonequilibrium states of model lattice systems,” Phys. Rev. B 28, 1655 (1983).

    ADS  Article  Google Scholar 

  10. S. Katz, J. L. Lebowitz, and H. Spohn, “Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors,” J. Stat. Phys. 34, 497–537 (1984).

    ADS  MathSciNet  Article  Google Scholar 

  11. K. Nagle, “Particle hopping models and traffic flow theory,” Phys. Rev. E 53, 4655 (1996).

    ADS  Article  Google Scholar 

  12. D. Helbing, “Traffic and related self-driven many-particle systems,” Rev. Mod. Phys. 73, 1067 (2001).

    ADS  MathSciNet  Article  Google Scholar 

  13. T. Tripathi and D. Chowdhury, “Interacting RNA polymerase motors on a DNA track: Effects of traffic congestion and intrinsic noise on RNA synthesis,” Phys. Rev. E. 77, 011921 (2008).

    ADS  Article  Google Scholar 

  14. S. Klumpp and R. Lipowsky, “Traffic of molecular motors through tube-like compartments,” J. Stat. Phys. 113, 233 (2003).

    MathSciNet  MATH  Article  Google Scholar 

  15. P. Greulich, A. Garai, K. Nishinari, A. Schadschneider, and D. Chowdhury, “Intracellular transport by single-headed kinesin KIF1A: Effects of single-motor mechanochemistry and steric interactions,” Phys. Rev. E 75, 041905 (2007).

    ADS  Article  Google Scholar 

  16. A. B. Kolomeisky, “Channel-facilitated molecular transport across membranes: Attraction, repulsion, and asymmetry,” Phys. Rev. Lett. 98, 048105 (2007).

    ADS  Article  Google Scholar 

  17. A. Zilman, J. Pearson, and G. Bel, “Effects of jamming on nonequilibrium transport times in nanochannels,” Phys. Rev. Lett. 103, 128103 (2009).

    ADS  Article  Google Scholar 

  18. P. Meakin, P. Ramanlal, L. M. Sander, and R. Ball, “Ballistic deposition on surfaces,” Phys. Rev. A 34, 5091 (1986).

    ADS  Article  Google Scholar 

  19. C. T. MacDonald, J. H. Gibbs, and A. C. Pipkin, “Kinetics of biopolymerization on nucleic acid templates,” Biopolymers 6, 1 (1968).

    Article  Google Scholar 

  20. F. Spitzer, “Interaction of Markov processes,” Adv. Math 5, 246–290 (1970).

    MathSciNet  MATH  Article  Google Scholar 

  21. B. Derrida, E. Domany, and D. Mukamel, “An exact solution of a one-dimensional asymmetric exclusion model with open boundaries,” J. Stat. Phys. 69, 66 (1992).

    MathSciNet  MATH  Article  Google Scholar 

  22. G. M. Schütz and E. Domany, “Phase transitions in an exactly soluble one-dimensional exclusion process,” J. Stat. Phys. 72, 277 (1993).

    ADS  MATH  Article  Google Scholar 

  23. B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, “Exact solution of a 1D asymmetric exclusion model using a matrix formulation,” J. Phys. A 26, 1493 (1993).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  24. H. Hinrichsen, “Matrix product ground states for exclusion processes with parallel dynamics,” J. Phys. A 29, 3659 (1996).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. A. Honecker and I. Peschel, “Matrix-product states for a one-dimensional lattice gas with parallel dynamics,” J. Stat. Phys. 88, 319 (1997).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. M. R. Evans, N. Rajewsky, and E. R. Speer, “Exact solution of a cellular automaton for traffic,” J. Stat. Phys. 95, 45 (1999).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  27. J. de Gier and B. Nienhuis, “Exact stationary state for an asymmetric exclusion process with fully parallel dynamics,” Phys. Rev. E 59, 4899 (1999).

    ADS  Article  Google Scholar 

  28. N. Rajewski, A. Schadschneider, and M. Schreckenberg, “The asymmetric exclusion model with sequential update,” J. Phys. 29, 305 (1996).

    ADS  MathSciNet  MATH  Google Scholar 

  29. N. Rajewski and M. Schreckenberg, “Exact results for one-dimensional cellular automata with different types of updates,” Physica A 245, 239 (1997).

    MathSciNet  Google Scholar 

  30. N. Rajewski, L. Santen, A. Schadschneider, and M. Schreckenberg, “The asymmetric exclusion process: Comparison of update procedures,” J. Stat. Phys. 92, 151 (1998).

    MathSciNet  MATH  Article  Google Scholar 

  31. N. Zh. Bunzarova and N. C. Pesheva, “One-dimensional irreversible aggregation with dynamics of a totally asymmetric simple exclusion process,” Phys. Rev. E 95, 052105 (2017).

    ADS  Article  Google Scholar 

  32. J. G. Brankov, N. Zh. Bunzarova, N. C. Pesheva, and V. B. Priezzhev, “A model of irreversible jam formation in dense traffic,” Physica A 494, 340 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  33. N. Zh. Bunzarova, N. C. Pesheva, V. B. Priezzhev, and J. G. Brankov, “A model of jam formation in congested traffic,” J. Phys.: Conf. Ser. 936, 012026 (2017).

    Google Scholar 

  34. N. Zh. Bunzarova, N. C. Pesheva, and J. G. Brankov, “One-dimensional discrete aggregation-fragmentation model,” Phys. Rev. E. 100, 0022145 (2019).

    ADS  Article  Google Scholar 

  35. M. Wölki, Master Thesis (Univ. Duisburg-Essen, Duisburg, 2005).

  36. A. E. Derbyshev, S. S. Poghosyan, A. M. Povolotsky, and V. B. Priezzhev, “The totally asymmetric exclusion process with generalized update,” J. Stat. Mech., 05014 (2012).

  37. A. E. Derbyshev, A. M. Povolotsky, and V. B. Priezzhev, “Emergence of jams in the generalized totally asymmetric simple exclusion process,” Phys. Rev. E 91, 022125 (2015).

    ADS  Article  Google Scholar 

  38. B. L. Aneva and J. G. Brankov, “Matrix-product ansatz for the totally asymmetric simple exclusion process with a generalized update on a ring,” Phys. Rev. E 94, 022138 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  39. P. Hrabák, Ph.D. Thesis (Czech Tech. Univ. Prague, Prague, 2014).

  40. P. Hrabák and M. Krbálek, “Time-headway distribution for periodic totally asymmetric exclusion process with various updates,” Phys. Lett. A 380, 1003–1011 (2016).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  41. N. C. Pesheva and N. Zh. Bunzarova, “gTASEP with attraction interaction on lattices with open boundaries,” arXiv:2001.02010v1, (2020).

  42. S. Janowsky and J. Lebowitz, “Finite size effects and shock fluctuations in the asymmetric simple exclusion process,” Phys. Rev. A 45, 618–625 (1992).

    ADS  Article  Google Scholar 

  43. S. Janowsky and J. Lebowitz, “Exact results for the asymmetric simple exclusion process with a blockage,” J. Stat. Phys. 77, 35–51 (1994).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  44. A. Jindal, T. Midha, and A. K. Gupta, “Analysis of interactions in totally asymmetric exclusion process with site-dependent hopping rates: Theory and simulations,” J. Phys. A 53, 235001 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  45. C. Bahadoran and T. Bodineau, “Properties and conjectures for the flux of TASEP with site disorder,” Braz. J. Probab. Stat. 29, 282–312 (2015).

    MathSciNet  MATH  Google Scholar 

  46. C. Arita, M. E. Foulaadvand, and L. Santen, “Signal optimization in urban transport: A totally asymmetric simple exclusion process with traffic lights,” Phys. Rev. E 95, 032108 (2017).

    ADS  Article  Google Scholar 

  47. A. Jindal, A. B. Kolomeisky, A. K. Gupta, The role of dynamic defects in transport of interacting molecular motors, Journal of Statistical Mechanics 2020. V. 2020 P. 043206.

  48. J. Brankov, N. Pesheva, and N. Bunzarova, “Totally asymmetric exclusion process on chains with a double-chain section in the middle: Computer simulations and a simple theory,” Phys. Rev. E 69, 066128 (2004).

    ADS  Article  Google Scholar 

  49. I. Neri, N. Kern, and A. Parmeggiani, “TASEP transport on networks: Theory for strongly connected networks,” Phys. Rev. Lett. 107, 068702 (2011).

    ADS  Article  Google Scholar 

  50. Y. Baek, M. Ha, and H. Jeong, “Effects of junctional correlations in the totally asymmetric simple exclusion process on random regular networks,” Phys. Rev. E 90, 062111 (2014).

    ADS  Article  Google Scholar 

  51. B. Pal and A. K. Gupta, “Role of interactions in a closed quenched system,” arXiv:2002.07075 (2020).

  52. L. Grüne, T. Kriecherbauer, and M. Margaliot, “Random attraction in the TASEP model,” arXiv:2001-07764 (2020).

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ACKNOWLEDGMENTS

The authors gratefully acknowledge joint work and fruitful discussions with their late colleagues and coauthors, Prof. V.B. Priezzhev and Prof. J.G. Brankov. Many results presented here are obtained in collaboration with them. This article is based on the presentation given at Memorial seminar dedicated to Vyacheslav Priezzhev. The authors very much appreciate the invitation and the support provided by the organizing committee of the Memorial seminar. Partial financial supports by the Bulgarian MES through Grant no. D01-221/03.12.2018 for NCDSC—part of the Bulgarian National Roadmap on RIs, and by the Plenipotentiary Representative of the Bulgarian Government at the Joint Institute for Nuclear Research, Dubna, through grant no. 01-3-1137-2019/2023 also are thankfully acknowledged.

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Bunzarova, N.Z., Pesheva, N.C. Aggregation-Fragmentation of Clusters in the Framework of gTASEP with Attraction Interaction. Phys. Part. Nuclei 52, 169–184 (2021). https://doi.org/10.1134/S1063779621020027

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  • DOI: https://doi.org/10.1134/S1063779621020027

Keywords:

  • nonequilibrium statistical mechanics
  • nonequilibrium stationary states
  • nonequilibrium phase transitions
  • traffic flow models
  • TASEP
  • aggregation–fragmentation of clusters
  • biological transport